Binary complexes and algebraic K-theory

Declaration of authorship......Page 7
Acknowledgements......Page 9
Introduction......Page 11
Notations and conventions......Page 15
1.1 Exact categories......Page 17
1.2 The Grothendieck group......Page 23
1.3 The higher K-theory of exact categories......Page 26
1.4 Chain complexes......Page 27
1.5 Binary complexes and Grayson's theorem......Page 36
2.1 Nenashev's relation and the shifting lemma......Page 39
2.2 K1 of a split exact category......Page 45
2.3 Some remarks on K2 and Milnor K-theory......Page 55
3 Fundamental theorems of K-theory via binary complexes......Page 59
3.1 The additivity theorem......Page 60
3.2 The resolution theorem......Page 63
3.3 The cofinality theorem......Page 69
4.1 Motivation: exterior powers and -rings......Page 77
4.2 The Dold–Kan correspondence and other preliminaries from homological algebra......Page 80
4.3 Operations on acyclic complexes......Page 84
4.4 Operations on binary multicomplexes......Page 88
4.5 Simplicial tensor products......Page 91
4.6 Exterior power operations on K-groups of schemes......Page 99
4.7 Making things explicit......Page 101
4.8 The -ring axioms for the higher K-groups of a scheme......Page 104
4.A Defining KnN with positive multicomplexes is harmless......Page 109
Bibliography......Page 111